Is the kind of job you apply for based on your pet's stats and if so what do you need to be able to be accepted for each job?

its just on intelligence of that pet. just try to read a lot of books to your pet and apply for a job you want. btw to get into SAI you need an intell. of about 500

So all the jobs are only based on intelligences?

yeah i guess, look at my pets: only the 2 basic colored ones dont have a job (or at leat one of them but i forgot wich)